On the Iwasawa λ-invariants of quaternion extensions by

ACTA ARITHMETICA LXXXVII.3 (1999)

On the Iwasawa λ-invariants of quaternion extensions

by

Keiichi Komatsu (Tokyo)

Dedicated to the memory of Prof. Dr. J¨urgen Neukirch

For a prime number l and a number field k, denote by λ_l(k) the Iwa- sawa λ-invariant associated with the ideal class group of the cyclotomic Zl-extension k∞(l) over k. It is conjectured that this invariant is zero for any prime l and any totally real number field k (cf. [7]). Several authors have given some sufficient conditions for the conjecture when k is a real abelian field (cf. [1]–[9]). Using them, many examples of the vanishing of λ-invariants for real abelian number fields are given. However, it seems that an example of a totally real non-abelian field has not yet been given. In this paper we give quaternion extensions K over the rational number field Q with λ₂(K) = 0. A Galois extension K over Q is called a quaternion extension if the Galois group G(K/Q) of K over Q is isomorphic to the quaternion group H₈ of order 8. The quaternion group H₈ is a group H₈ = hσ, τ i of order 8 with σ⁴= 1, σ²= τ² and τ στ⁻¹ = σ⁻¹.

The main purpose of this paper is to prove the following:

Theorem. Let p be a prime number with p ≡ 3 (mod 8), k = Q(√ 2,√

p) and k∞(2) the cyclotomic Z2-extension of k. Then there exist natural num- bers x, y with x²− y²p = 2p. Let K_∞(2) be the cyclotomic Z₂-extension of K = k(

q

(x + y√

p)(2 +√

2)). Then the Galois group G(K/Q) of K over Q is isomorphic to the quaternion group H₈ and the λ-invariant λ(K_∞(2)/K) of K∞(2) over K vanishes.

First we recall the following lemma which plays an important role in our proof of this theorem:

Lemma (cf. [2]). Let l be a prime number , k a totally real number field of finite degree and K a real cyclic extension of degree l over k. Assume that

1991 Mathematics Subject Classification: Primary 11R23.

[219]

220 K. Komatsu

k∞(l) has only one prime ideal lying over l and that the class number hk of k is not divisible by l. Then the following are equivalent:

(1) λ(K_∞(l)/K) = 0.

(2) For any prime ideal w of K_∞(l) which is prime to l and ramified in K_∞(l)/k_∞(l), the order of the ideal class of w is prime to l.

P r o o f (of Theorem). Since p ≡ 3 (mod 8), we have N_Q(^√_p)/Q(Q(√ p)^×) 63 −1. Hence the cardinality of the ambiguous classes of Q(√

p) is equal to one, which shows that a prime ideal of Q(√

p) lying above 2 is principal.

Therefore there exist integers x, y with x²− py²= 2p by ⁻²_p

= 1. We put α =

q (2 +√

2)(x + y√

p). Now, let σ, τ be elements of the Galois group G(k/Q) with √

2^σ = −√ 2, √

p^σ =√ p,√

2^τ =√

2 and √

p^τ = −√

p. Then we have (α²)(α²)^σ = 2(x + y√

p)² and (α²)(α²)^τ = 2p(2 +√

2)², which shows that K is a Galois extension over Q. For simplicity, we denote by σ, τ extensions of σ, τ to K with α^σ=√

2α⁻¹(x + y√

p) and α^τ =√

2pα⁻¹(2 +

√2). Then we can easily see G(K/Q) = hσ, τ i, σ⁴= 1, σ²= τ²and τ στ⁻¹ = σ⁻¹. Hence G(K/Q) is isomorphic to H8.

Now, we prove λ(K∞(2)/K) = 0. First we notice that the class number h_Q(^√_p)is not divisible by 2. Therefore h_kis not divisible by 2, since 2 is fully ramified in k over Q and since p is unramified in k over Q(√

p). One should also remark that the infinite primes are unramified. Let Ppbe a prime ideal of K lying above p, p₂ a prime ideal of k lying above 2 and p_pa prime ideal of k lying above p. Then we can see ((x + y√

p)(2 +√

2)) = p²_pp²₂. Hence we have (α) = Pp(2 +√

2) in K. This shows that Pp is a principal ideal of K.

Therefore λ(K_∞(2)/K) = 0 follows from the Lemma or [7, Lemma 3]. This completes our proof.

Remark. Since there exist infinitely many prime numbers p with p ≡ 3 (mod 8) which are unramified in k/Q(√

p), there exist infinitely many quaternion extensions K with λ(K_∞(2)/K) = 0.

References

[1] T. F u k u d a and K. K o m a t s u, On Zp-extensions of real quadratic fields, J. Math.

Soc. Japan 38 (1986), 95–102.

[2] T. F u k u d a, K. K o m a t s u, M. O z a k i and H. T a y a, Iwasawa λp-invariants of relative cyclic extensions of degree p, Tokyo J. Math. 20 (1997), 475–480.

[3] T. F u k u d a and H. T a y a, The Iwasawa λ-invariants of Zp-extensions of real quad- ratic fields, Acta Arith. 69 (1995), 277–292.

[4] R. G r e e n b e r g, On the Iwasawa invariants of totally real number fields, Amer. J.

Math. 98 (1976), 263–284.

Iwasawa λ-invariants of quaternion extensions 221

[5] H. I c h i m u r a and H. S u m i d a, On the Iwasawa λ-invariants of certain real abelian fields, to appear.

[6] —, —, On the Iwasawa λ-invariants of certain real abelian fields II , to appear.

[7] K. I w a s a w a, A note on capitulation problem for number fields II , Proc. Japan Acad.

65 (1989), 183–186.

[8] J. S. K r a f t and R. S c h o o f, Computing Iwasawa modules of real quadratic number fields, Compositio Math. 97 (1995), 135–155.

[9] M. O z a k i and H. T a y a, On the Iwasawa λ₂-invariants of certain families of real quadratic fields, submitted for publication (1996).

Department of Information and Computer Science School of Science and Engineering

Wasada University

3-4-1 Okubo, Shinjuku, Tokyo 169-8555 Japan

E-mail: kkomatsu@mse.waseda.ac.jp

Received on 22.5.1997

and in revised form on 21.9.1998 (3193)